2 HUGH THOMAS Theorem

We show that a certain orbit category considered by Keller encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows us to give type-uniform proofs of certain results of Fomin and Reading in the simply laced cases. For Φ any root system, Fomin and Zelevinsky [FZ] define a cluster complex ∆(Φ), a simplicial complex on Φ ≥−1 , the almost positive roots of Φ. Its facets (maximal faces) are called clusters. In [BM+], starting in the more general context of a finite dimensional hereditary algebra H over a field K, Buan et al. define a cluster category C(H) = D b (H)/τ −1 [1]. (D b (H) is the bounded derived category of representations of H; more will be said below about it, its shift functor [1], and its Auslander-Reiten translate τ .) The cluster category C(H) is a triangulated Krull-Schmidt category. We will be mainly interested in the case where H is a path algebra associated to the simply laced root system Φ, in which case we write C(Φ) for C(H). There is a bijection V taking Φ ≥−1 to the indecomposables of C(Φ). A (cluster)-tilting set in C(Φ) is a maximal set S of indecomposables such that Ext 1 C(Φ) (X, Y) = 0 for all X, Y ∈ S. C(Φ) encodes the combinatorics of ∆(Φ) in the sense that the clusters of Φ correspond bijectively to the tilting sets of C(Φ) under the map V. Tilting sets in C(Φ) always have cardinality n, the rank of Φ. An almost complete tilting set is a set T of n − 1 indecomposables such that Ext 1 C(Φ) (X, Y) = 0 for X, Y ∈ T. A complement for T is an indecomposable M such that T ∪ {M } is a tilting set. A tilting set always has exactly two complements. (This was shown from the cluster perspective in [FZ] and from the representation theoretic perspective in [BM+].) In [FR], Fomin and Reading introduced a generalization of clusters known as m-clusters, for m ∈ N. When m = 1, the classical clusters are recovered. The m-cluster complex ∆ m (Φ) is a simplicial complex on a set of coloured roots Φ m ≥−1. It has been studied further …